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Unformatted text preview: MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering 2.004 Dynamics and Control II Fall 2007 Problem Set #9 Solution Posted: Sunday, Dec. 2, 07 1. The 2.004 Tower system. The system parameters are m 1 = 5 . 11kg, b 1 = 0 . 767N sec / m, k 1 = 2024 N / m; m 2 = 0 . 87 kg, b 1 = 8 . 9 N sec / m, k 1 = 185 N / m. The system model schematic is shown again below for your convenience. Eigenvalues and Eigenvectors in state space representation The eigenvalues of A matrix are poles. Hence, the imaginary parts of the poles are the damped oscillation frequencies ( d = n 1 2 ), and the real parts are n . The eigenvectors of A matrix represent the modes of the system. Each element of a certain eigenvector indicates the way that state variables change when the system is excited with the corresponding damped natural frequency. For example, if an eigenvector is [ 2 1] T , then the two state variables move in opposite way and the ratio of their magnitudes is 2. The real and imaginary parts of the eigenvalues represent decay rate (or settling time) and natural frequency. For the complex eigenvector, you may interpret that the real and imaginary part correspond to cos and sin components of the response. Since cos = sin( + / 2) and exp { j/ 2 } = j , the real part represents the magnitude of cos( t ) motion, and the imaginary part represents the magnitude of sin( t ) motion. To give you an easy example, lets consider the 2.004 Tower system without damping. Then the system matrix is 1 A = 432 . 2896 36 . 2035 1 , 212 . 6437 212 . 6437 the eigenvectors are j . 0354 j . 0354 j . 0106 j . 0106 . 7614 . 7614 . 1427 . 1427 v 1 = , v 2 = , v 3 = , v 4 = . j . 03 j . 03 j . 0732 j . 0732 . 6466 . 6466 . 9870 . 9870 1 and the eigenvalues are { j 21 . 5183 , j 13 . 4870 } . Note that the eigenvalues are purely imaginary because there is no damping. Lets look at v 3 corresponding to the lower natural frequency 13 . 4870 (rad/s). The first ( j . 0106) and third ( j . 0732) elements are the displacement of the tower and slider, respectively. So they move along the same direction in sin fashion (because they are imaginary) without phase delay. The second (0 . 1427) and fourth (0 . 9870) elements are velocity of the tower and the slider, respectively. They change with the same sign in cos fashion (because they are real) without phase delay. Since velocity is a time derivative of displacement, it totally makes sense that the first and third elements are purely imaginary while the second and fourth ones d are pure real, because (sin t ) cos( t )....
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This note was uploaded on 02/23/2012 for the course MECHANICAL 2.004 taught by Professor Derekrowell during the Fall '08 term at MIT.
 Fall '08
 DerekRowell
 Mechanical Engineering

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